Strongly isotropy irreducible spaces whose Laplacians have all their eigenspaces irreducible

Abstract

Let $H$ be a closed subgroup of a compact connected Lie group $G$ such that $G/H$ is simply connected, the action of $G$ on $G/H$ is almost effective, and $G/H$ is a strongly isotropy irreducible space. The smallest positive eigenvalue of the Laplace-Beltrami operator $\Delta$ associated to the standard metric on $G/H$ was recently computed. In this article we obtain a second non-zero eigenvalue of $\Delta$. As a consequence, we prove that all eigenspaces of the Laplacian are irreducible as $G$-modules if and only if $(G,H)$ is a compact symmetric pair of real rank one, or $(G,H)$ is equal to $(G_2,\mathrm{SU}(3))$ or $(\mathrm{Spin}(7),G_2)$.